Expression of type Forall ¶

from the theory of proveit.numbers.multiplication ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Function, Q, f
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Mult, Sum
from proveit.numbers.summation import summation_b1toj_fQ

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(f, sub_expr1)
sub_expr3 = Function(Q, sub_expr1)
expr = Forall(instance_param_or_params = [f, Q], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, Complex), condition = sub_expr3), Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, summation_b1toj_fQ, c_1_to_k), Sum(index_or_indices = sub_expr1, summand = Mult(a_1_to_i, sub_expr2, c_1_to_k), condition = sub_expr3)).with_wrapping_at(2), domain = Complex).with_wrapping()).with_wrapping_at(2))

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{f, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}\right)\end{array}\right] \end{array} \end{array}\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	56, 53
4	Operation	operator: 5 operands: 6
5	Literal
6	ExprTuple	7, 8
7	Operation	operator: 10 operand: 12
8	Operation	operator: 10 operand: 13
9	ExprTuple	12
10	Literal
11	ExprTuple	13
12	Lambda	parameters: 57 body: 14
13	Lambda	parameters: 15 body: 16
14	Conditional	value: 17 condition: 49
15	ExprTuple	50, 52
16	Conditional	value: 18 condition: 19
17	Operation	operator: 41 operands: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	ExprTuple	51, 45
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 47 operands: 29
26	Operation	operator: 37 operand: 34
27	ExprRange	lambda_map: 31 start_index: 65 end_index: 55
28	ExprRange	lambda_map: 32 start_index: 65 end_index: 59
29	ExprTuple	50, 33, 52
30	ExprTuple	34
31	Lambda	parameter: 71 body: 35
32	Lambda	parameter: 71 body: 36
33	Operation	operator: 37 operand: 43
34	Lambda	parameters: 57 body: 39
35	Operation	operator: 41 operands: 40
36	Operation	operator: 41 operands: 42
37	Literal
38	ExprTuple	43
39	Conditional	value: 44 condition: 49
40	ExprTuple	60, 45
41	Literal
42	ExprTuple	62, 45
43	Lambda	parameters: 57 body: 46
44	Operation	operator: 47 operands: 48
45	Literal
46	Conditional	value: 51 condition: 49
47	Literal
48	ExprTuple	50, 51, 52
49	Operation	operator: 53 operands: 57
50	ExprRange	lambda_map: 54 start_index: 65 end_index: 55
51	Operation	operator: 56 operands: 57
52	ExprRange	lambda_map: 58 start_index: 65 end_index: 59
53	Variable
54	Lambda	parameter: 71 body: 60
55	Variable
56	Variable
57	ExprTuple	61
58	Lambda	parameter: 71 body: 62
59	Variable
60	IndexedVar	variable: 63 index: 71
61	ExprRange	lambda_map: 64 start_index: 65 end_index: 66
62	IndexedVar	variable: 67 index: 71
63	Variable
64	Lambda	parameter: 71 body: 68
65	Literal
66	Variable
67	Variable
68	IndexedVar	variable: 69 index: 71
69	Variable
70	ExprTuple	71
71	Variable

Expression of type Forall¶

from the theory of proveit.numbers.multiplication¶

Expression of type Forall ¶

from the theory of proveit.numbers.multiplication ¶